Divisibility properties of sum of powers of successive natural numbers and Bowen’s hypothesis

Abstract

The explicit formulae for remainder of sum of powers of successive natural numbers (SPSN) divided by any prime factor p of the number N of sum terms is derived. The remainder is zero, if N contains pj, j>1. It is also 0, if j=1 and the exponent is not divisible by p-1. The expression (N-N/p) (mod p), named “eigen-remainder” introduced for N is the remainder of SPSN in the rest of cases. All natural numbers, having all eigen-remainders equal to 1, are found not exceeding 42. It is demonstrated that in all other cases, there exists a prime divisor of N, which dividing SPSN gives the remainder ≠ 1. This proves Bowen’s hypothesis.